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Q: When I view the surface of cooking oil in a pan by reflected light, a pattern of honeycomb-like shapes appears as the pan is heated by a gas flame. The unit size of the pattern is smallest where the oil is thinnest. Why is this? (continued)

Readers have already provided answers to this question. However: as our correspondent below points out, the previous explanations using the Rayleigh convection model were not wholly correct, for the Rayleigh model only applies if the frying liquid is of sufficient depth – Ed

A: The behaviour of hot oil in a pan is a classic example of Bénard convection, the unstable motion of fluid on a heated flat plate which takes the form of regular hexagonal cells of circulating fluid. It is well known that Lord Rayleigh developed a theory to explain this instability. What is not so well known is that his model was wrong.

Rayleigh considered a horizontal layer of liquid with flat surfaces heated from below, and assumed that the instability took the form of parallel, contra-rotating rolls driven by buoyancy forces due to variations in the fluid density. Then, by heuristic arguments, he deduced a size for hexagonal cells close – fortuitously – to that observed by Bénard. He also predicted the minimum temperature gradient across the layer for the onset of this motion but this was about 100 times greater than the gradient needed to initiate the cellular flow in Bénard’s experiments.

Other researchers extended Rayleigh’s analysis in various ways. When the flat upper surface condition was later relaxed it could be seen that the surface is elevated above rising fluid between adjacent rolls while it is depressed above descending fluid. This is precisely the opposite of what Bénard observed. When Bénard’s experiment was repeated it was found that the cells could also be produced when the heating plate was cooled, whereas according to Rayleigh’s ideas the fluid should remain at rest. The instability has also been observed for a layer of liquid beneath a plate being heated from above and in space, where gravity and hence buoyancy forces are zero.

In the late 1950s a new model for Bénard convection was developed in which variations of surface tension caused by temperature variations on the surface of the liquid drove the motion. This model also predicted a depressed surface above rising fluid. In reality both Bénard and Rayleigh effects must be present. Conditions determine which predominates. Buoyancy forces drive the motion when there is no free surface or the liquid layer is thicker than about 10 millimetres; otherwise surface tension governs the flow.

Whichever driving force dominates, it must be sufficient to overcome the effects of viscous drag (which tends to inhibit motion) and diffusion of heat within the fluid (which tends to reduce the temperature gradients) before it can initiate the unstable flow. For buoyancy-driven flows the onset of instability is governed by the Rayleigh number: buoyancy forces/(viscous drag × rate of heat transfer)

while for flows driven by surface tension the corresponding variable is the Marangoni number in which surface tension forces replace buoyancy forces.

For thin layers the unstable flow takes the form of a regular array of hexagonal cells regardless of the shape of the container. For thicker layers the basic unstable flow is a series of rolls parallel to the container’s sides with the direction of flow adjacent to its rim and determined by its temperature relative to its base. These rolls degenerate into polygonal (but not necessarily hexagonal) cells when the temperature gradient is increased.